As director of UMD’s applied mathematics & statistics, and scientific computation graduate program, Radu Balan oversees many branches of math—and much more.

Radu Balan doesn’t consider himself a mathematician, which might sound surprising coming from a University of Maryland Professor of Mathematics.
“I’m an engineer at my core,” Balan said. “In fact, my background is engineering first, then physics.”
With a Ph.D. in applied and computational mathematics to boot, Balan’s interdisciplinary background serves him well in a leadership role he assumed in July 2024: director of UMD’s applied mathematics & statistics, and scientific computation (AMSC) graduate program. 

“Our 150-plus affiliated faculty come from the entire campus and all sorts of research areas, which is one of our selling points,” Balan said. “We like to say that AMSC is a structured program with flexibility.”

The flexibility to pursue different academic disciplines has always been important to Balan, who wasn’t exposed to applied mathematics until later in life.

Finding mathematical harmony

Balan grew up in Romania, a country with a strong mathematical tradition, but initially took an interest in other subjects. He earned a bachelor’s degree in electrical engineering from the Polytechnic Institute of Bucharest in 1992, followed by a bachelor’s in physics from the University of Bucharest in 1994. At some point along the way, he also discovered applied math.

Inspired by the prospect of applying math to fields he already enjoyed, Balan enrolled in Princeton University’s applied and computational mathematics Ph.D. program. His advisor, Ingrid Daubechies, has been dubbed the “godmother of the digital image” for her pioneering work on wavelets that led to image-compression technologies like JPEG 2000, an improvement to the original JPEG image file format.

Daubechies introduced Balan to the field he currently specializes in: harmonic analysis, which breaks functions down to the sum of their parts, such as waves and signals, and has broad applications in medical imaging, audio processing, machine learning and more. Above all else, Daubechies gave Balan an entirely new perspective on mathematics.

“Ingrid shaped not only my career but my vision of applied math,” Balan said. “I learned to have an open mind in math and not live in an ivory tower and say, ‘This is the truth, and therefore this is the only problem I need to work on.’”

After graduating in 1998, Balan was eager to put everything he had learned into practice and accepted a position as a research scientist at Siemens, where he studied signal processing and communications for eight years.

Though Balan enjoyed his time there, his goal was to work in academia. The opportunity arose in 2007 when he accepted a position as an associate professor at UMD. Balan said he was attracted to UMD’s Department of Mathematics for its active research, including advancements in harmonic analysis. 

Balan now serves as co-director of the Norbert Wiener Center for Harmonic Analysis and Applications and is an affiliate with the Artificial Intelligence Interdisciplinary Institute at Maryland. 

Papers and patents

During his career, Balan has published nearly 50 papers in academic journals, including a mix of mathematical theory and applications.

Balan is best known for his work on phase retrieval, which uses algorithms to reconstruct lost information about the timing and position of an electromagnetic wave, sound wave or other signal. While at Siemens, Balan’s 2005 signal reconstruction research led to the invention of a device to help solve what’s known as the “cocktail party problem.”

“At a cocktail party, you're in a room full of people talking more or less simultaneously, so we created a device with microphones to pick up what individual speakers are saying,” Balan explained. “It reduces interference and helps you focus on one speaker at a time.”

Balan has contributed to more than 20 patented inventions—mostly at Siemens, but one while working at UMD in 2017. The latter was a device that used a microphone network to improve the monitoring of industrial equipment.

“You have these sensors—microphones mostly—to monitor each active component for abnormalities or something breaking down,” Balan explained.

Aside from his research, Balan is also one of three editors-in-chief for the Applied and Computational Harmonic Analysis journal and serves on the editorial board of the American Institute of Mathematical Sciences, which oversees several journals.

Balan also monitors the current state of applied math and the job market to ensure AMSC students gain the most valuable skills for applied mathematicians.
“What I enjoy most about my job is discovering new problems,” he said, “but more importantly, new solutions.”

Written by Emily Nunez

Math Professor Reimagining Learning: Meet Qendrim Gashi

Let’s start by talking about your role in the field of mathematics. Can you tell me what inspired you to pursue a career in mathematics and your specific concentration areas?“It was the beauty, the universality, and the challenge of mathematics that attracted me to it. Also, the passion of my teachers when I was 11-12, [and] the mystery that sometimes surrounds mathematics. People often are in awe when you say that you are a mathematician or that you study mathematics. Plus there was a local math olympiad that was being organized. That made me want to advance in doing [and] reading more mathematics [as well as] solving more mathematical problems.
 

"My specialty, or specific concentration area, is related to combinatorial study of Representation Theory, the study of symmetries, and of algebraic and arithmetic algebraic geometry. That is the study of geometry using algebra and algebraic tools, and when you add arithmetic to it it means problems arising from number theory. That is my research focus, and the idea is to try and recognize patterns [as well as] collect objects because you have Combinatorics behind. One can say that I am a pure mathematician.”
 

Is pure mathematics more difficult than other types of mathematics?“Not necessarily. For me, applied mathematics is more difficult. Pure mathematics, in this case, means that you are looking at problems purely from a mathematical point of view and you are studying them without necessarily looking for models to apply them to, outside of mathematics. But in reality, there is a unity of mathematics and even those that do pure mathematics end up contributing to applications of mathematics. Then there are some who do applied mathematics who advance the theory of pure mathematics. The two are two parts of the same [whole].”
"Can I make a comment on people who say they are 'not very good at math?'

“Often, when I meet people, and they find out that I am a mathematician there are two strong reactions. One is of the people who say “I’m not good at math!” and they are thankful that they have made it so far without mathematics. There is also the class of those that says ‘Oh I was always excellent at math,’ ‘I could have actually become a mathematician if I wanted to.’ Part of this is related to the fact that mathematics is generating strong emotions among people because mathematics is often used as a tool to rank students: at a university, at a high school, at other levels. I think that we need to change that because mathematics is not about ranking students. It should be a way of thinking that enriches everyone’s lives.

"Let’s say I am a writer. I would assume that the probability is very high that the person speaking with me is not the future Nobel Prize winner in Literature. [But] that doesn’t somehow generate these strong emotions among people [the same] as when you say that you’re a mathematician. This, I think, is something that we as a mathematical community need to address, and society as a whole. Because we lose many people, who just shut down their brains, because they say ‘math is not for me.’ While it’s true that you can thrive and have a very good life without mathematics, you can say the same thing about having a good life without knowing who Shakespeare was. But it would be a life which would not be as rich and as colorful.

"It is unfortunate that we present mathematics as something that is unique to all those ‘bright kids.’ There are many bright kids who are not mathematicians, and statistically many more [who are not mathematicians]. There is work somewhere that we need to do. The reality is that this is a problem, a challenge, everywhere around the world.”

I noticed you took this passion for math and formed the Kosovar Mathematical Society in 2008. Can you tell me more about this?

“In 2008, I obtained my Ph.D. from the University of Chicago and I was planning to go back to Kosovo. I wanted to do something to give back to the community that had helped me. Also, because I lived in Kosovo during a particularly challenging period during the 90s, I thought that it would be important to give back to the community. The direct contribution that I thought I could make was by creating a local math society, which I did together with a number of colleagues. We managed to then organize seminars, conferences, and a local version of the Math Olympiad. Then our participants could take part in the International Math Olympiad. That was important because, this way, many of my former students went through this path and some went to universities around Europe. Some even came to the U.S.

“For me, it was about using something that I had. Namely this experience, but also I had [become] somewhat famous in Kosovo and thought I should use my fame for something positive, and [to] extend this passion to a new generation. I am delighted that among these participants, these students, I have many who are extremely bright and talented. The challenge is how to keep that community together, because as with many countries around the world, there is this 'brain drain.' A community that stays together and that contributes to science [is not so easy to maintain] in a smaller country that doesn’t necessarily have the means. That is the next step, but I am very proud that I have some brilliant students who are doing very well.”

Can you explain what the International Math Olympiad is and why it’s important for mathematicians from Kosovo to attend?

“This is the most famous math competition for high school students [and] has been going on for decades now. [However], there is a tendency to reduce pre-university education math to these competitions, which is also wrong. This [competition] is one of the tools available to promote mathematics, to promote discussion among young mathematicians [and] young high school students, who are interested in mathematics. [As well as] to make sure local communities, in this case the math community in Kosovo, [are] connected and communicate with their peers around the world.

"This is what the International Math Olympiad is, but this is only one vector of operation of the Math Society that I created. Incidentally, we were supported by many mathematicians from around the world - it was not just a local initiative! It was…but I was helped by many international actors.

"Math is both a universal language and it can cause universal confusion. We [mathematicians] are here to make sure that the latter doesn’t happen too much.”

As a math professor, do you have any specific goals in educating the next generation of mathematicians? What do you hope to see for the future of mathematics?

“I attempt to approach it a little bit like a theater play: where it’s true there’s a light that shines over the stage - but the point is that you are trying to enlighten the audience. I just hope it’s not a Greek Tragedy when I teach! But my aim is to make sure students appreciate the mathematical thinking and structures, that they are able to then become critical thinkers, use the tools of mathematics to apply them in various walks of life, and of course the passion. I hope [this passion] is contagious, where they take whatever they want to do and move it one step further - advance it.

“So if they want to become mathematicians, that’s great! But they don’t have to become mathematicians. They can use mathematics in many professions. For me, when I am in front of my students, I try to also understand them and this is part of the challenge: making sure that what you are saying is relevant, not just to the syllabus but also making sure that [it] becomes relevant to their education. In the end, there’s this ‘noble’ idea of educating the future generation of those that will change the world. I am happy if I’m part of that effort. [However] in mathematics, you have to convey the material, some messages, and also this passion [for math].”

What level courses are you teaching here at the University of Maryland?

“I am teaching two [undergraduate] courses. One is 'Introduction to Calculus,' a 100-level course. The other is a 400-level course, mostly for math majors. It’s a course on groups, rings, and fields. This is abstract algebra. They are two different experiences. One course is a larger course with students who are not always thrilled to have another math midterm, but who have been very good about working hard. I am very happy that they’re taking my course. Then another smaller group of students who have already decided that they want to do math or will use math in their fields and so now they are learning more advanced notions.

“I’ve taught many courses and I like the challenge of both going in front of a somewhat skeptical audience and also in front of a hungry audience. That’s why I made the comparison [earlier] with theater play and a performance. You have to really give a lot of energy to make sure the messages go through. But I have been very positively impressed about what I have seen so far among the students here.

“The [benefit of] 'Universality of Mathematics' is its application in many fields. That’s why it’s a required course!”

You also held several respectable diplomatic positions between Kosovo and France in your lifetime. Does this have any overlap with your knowledge of mathematics, for those of us who may not understand diplomacy roles well?

“I think people are more inclined to understand what diplomats do than what mathematicians do! But in my case, I had this opportunity to serve as an ambassador of Kosovo to France and also, during a period, to Portugal, Monaco, and Andorra, covering from Paris as a non-resident Ambassador. That was something that I was asked to do by the state authorities in Kosovo. I was already promoting the Francophonie, the French Language, in Kosovo. I was also fairly active in the political life in Kosovo, [so] it seemed very natural to accept this. It was an exceptional experience. Of course, mathematics doesn’t overlap in the classic sense because you’re not trying to do research about a diplomatic mission, but it is about advancing knowledge and understanding: trying to understand the other parties, trying to forge relations, trying to understand the quality of those relations, giving context and understanding context, and building networks.

“All of this is something that is useful in academia. I think that in the current age, we live practically under a period where mathematics and mathematical ideas have revolutionized our way of life. It is essential to have people who understand various parts of [the] human endeavor, including sciences and mathematics, contribute equally to other parts of human endeavor. It should not be a complete surprise if you have someone who has a background in sciences doing diplomacy because that adds to the spectrum of people with certain skills who can transfer them in their discussions. These days, in our digital era, diplomacy is essential to understand a bit better how people function. There is space for more scientists to be included in [diplomacy]. It is true that mathematicians have a reputation of being socially awkward, but I think a little bit of champagne helps!”

How has being a diplomat between Kosovo and France [and the other nations has] positively impacted you as an international scholar?

“One of the reasons why I like modern academic environments is because you get to meet people from different corners of the world who have different backgrounds and different viewpoints. You learn from them, even if you disagree with them. [I think] that is fundamental to an environment that is open and promotes academic freedom. I was lucky to have been exposed to this throughout my education and professional career. [I also had] this phase where I was a diplomat at a very big and important capital, [which I hope] has contributed to opening up my horizons and further forging these networks of cooperation. I think it has also made me appreciate the conversations with others [and] the necessity to share and learn more from one another. Because of the way academic life is structured, teaching, research… you often don’t have enough time to converse with other people. Being in an environment where conversation was essential, for me, was an interesting and positive experience.

 “I might add one thing: I was very lucky to be posted in France where I have many friends and where I was accepted. I appreciate, and always tried to learn more about the French language and culture. I am now proud to have, in my family, kids who are French speaking as well.”

Are there any experiences you credit specifically to your area of study, pure mathematics?

“Apart from this universality of mathematics, which is extremely important, when discussing with people we’re transcending cultures and all sorts of differences. One thing that has had an impact was the idea that ‘I am contributing to constructing knowledge and advancing human-kind by adding very little, minuscule contributions…but in an irreversible way.’ This is something that you don’t necessarily get [as] a diplomat. You can contribute to having some agreement or deal between two countries, but that agreement could end up not being applied or [ending] in a few years time. Whereas with mathematics, if you are proving something that is true, you are advancing human knowledge. I find that very beautiful and satisfying. That’s something that I like [about mathematics]; the idea that what I have been studying lies at the intersection of a number of mathematical fields. You’re always trying to understand things in a number of domains and trying to compare them, see the relations between them, find patterns that are common…that is the type of problem I like to address when I’m thinking about research. This is something that has become a part of my intellectual existence: both curiosity and a desire to understand things. That’s what I hope to transmit to my students and future generations.”

Is there anything really specific that is credited to you in mathematics that you feel has started this push to transmit future knowledge?

“I’m only a miniscule contributor in a number of fields, but I like challenges. There’s this classic thinking that there are two types of mathematicians: those that solve problems and those that create theories. I would probably belong more to the first category. I am happy that I have solved some problems, but I would like to solve more! At some point, it becomes not just about solving problems but about understanding them. If you solve them, then you’re happy! But I have worked on a number of problems related to Combinatorial Aspects of Representation Theory, where I hope I have left a little bit of a mark [despite] the giant mathematicians working in this field.”

Do you have any advice for younger people who may be pursuing either of these career paths?

“I hope that younger mathematicians will carry their torch and enlighten [the] future generations. In particular, [I hope they will] contribute to the understanding and advancement of mathematics, to its use in society… but at the same time, think about how they can contribute to a more peaceful, more prosperous world where there’s more understanding between individuals, nations, and communities. That’s my wish and hope [for the future].

“I would like to add that I am delighted that currently I am able to both contribute to UMD and its students and also learn from them. In particular, I am grateful for the very warm welcome that I received at the very active and distinguished Math Department here at UMD.”

For more than two decades, Zbigniew Błocki has been sharing the story of the Polish codebreakers who helped crack Nazi Germany’s Enigma machine—with math.

What was the turning point that led the Allies to victory in World War II? Some might cite a particular battle or army, but University of Maryland Mathematics Visiting Professor Zbigniew Błocki offered another perspective during a Department of Mathematics colloquium in February.

“Some people say that mathematicians won the Second World War,” said Błocki, a mathematics professor from Jagiellonian University in Krakow, Poland. “Maybe that is exaggerated, but I would argue not that much.”

Periodically over the last 24 years, Błocki has given talks on the lesser-publicized role that Polish mathematicians played in decrypting Enigma, the typewriter-like machine that Nazi Germany used to conceal its communications. English mathematician Alan Turing is widely credited for cracking the code, but few know that Marian Rejewski from Poland’s Cipher Bureau laid the foundation.

“When it comes to Enigma, people outside of Poland have usually heard about Alan Turing but rarely about Marian Rejewski,” Błocki said. “I am still surprised how little in general people know about the story.”

A cryptologic bomb

For decades after World War II ended, Rejewski’s role in cracking Enigma—along with the contributions of his colleagues Jerzy Różycki and Henryk Zygalski—was classified. This information wasn’t made public until the 1970s, and Błocki only learned about it in the mid-1980s while reading a book on the subject.
As a lover of math and history, Błocki was impressed to learn how these cryptologists used a mix of math, mainly permutation group theory, and human psychology to capitalize on the Germans’ mistakes. At the time, Enigma was considered virtually unbreakable because of its complex network of rotating wheels and plugboard connections that scrambled the letters each time a key was pressed.

However, these encrypted messages followed a predictable format, providing the Polish mathematicians with clues. By December 1932, Rejewski’s team had leveraged crucial intelligence obtained by the French to figure out Enigma’s internal wiring pattern. When coding procedures were overhauled in September 1938, making their decoding efforts much harder, the Poles built a device dubbed a “bomba,” or bomb, that automated the process of determining Enigma’s daily settings. 

Coupled with earlier developments that allowed the Poles to keep pace with German improvements, the bomba—an early precursor to computing—enabled the Polish to read intercepted German communications from 1933 to 1939. After handing these findings off to the British, Turing and mathematician Gordon Welchman designed the Bombe machine that greatly advanced the Allies’ decryption efforts and helped them win key battles.
“The breaking of the Enigma code had an incredible impact on the outcome of the Second World War,” Błocki said. “It was important everywhere the Allies were fighting the Germans, including the Battle of Britain, where, thanks to decoded Enigma messages, the British knew quite in advance where the attacks would be.”

Setting the record straight

Błocki noted that despite garnering some international attention, Poland’s role in cracking the Enigma code remained a footnote in history for many years—or worse, the subject of misinformation.

“Twenty years ago, the item on Enigma in Encyclopedia Britannica didn't mention Poles at all,” Błocki said. “The problem culminated in 2014 with the Hollywood blockbuster on Alan Turing, ‘The Imitation Game’ starring Benedict Cumberbatch. In it, the role of Polish cryptologists was again missing and the totally false story of a Pole who stole an Enigma machine from the Germans was repeated.”

Since then, Błocki has been committed to setting the record straight. The first time he gave his talk on Enigma was for his habilitation defense—a postdoctoral presentation on a subject separate from one’s research—at Jagiellonian University in 2001.

“It was very crowded with well over 100 people,” Blocki said of his audience. “The atmosphere was pretty patriotic, and by the end of it, they seemed almost ready to sing the Polish national anthem.”

Błocki has since adapted his presentation for a range of audiences, including scientific institutions, universities, high schools and even his son’s elementary school class. When he left his position with Poland’s National Science Centre, where he served as director from 2015 to 2023, Błocki received a unique parting gift from his colleagues: an electronic copy of Enigma. 

After joining UMD, Błocki shared the story of Enigma with Mathematics Professor Yanir Rubinstein, who encouraged Błocki to present the topic at a department colloquium. While reflecting on the story’s reach since it was uncovered in the ’70s, Błocki feels encouraged by recent efforts to portray this chapter of history more accurately.
“Fortunately, a lot has changed. For example, the Encyclopedia Britannica entry about Enigma has been completely revised,” he said. “I think the 2018 publication of ‘X, Y & Z: The Real Story of How Enigma Was Broken,’ a book by Alan Turning’s nephew, Dermot Turing, was very important. This is probably the first British publication of its kind where the critical role of Polish mathematicians in this story is presented.”

For Błocki, this lecture is also a fun way to teach mathematical and computing concepts through history.
“I gave this talk at the University of Bonn in Germany in 2001, and the professor who organized it said he had been to several talks on Enigma, but mine was the only one that explained the mathematics,” Błocki said. “Although from today's viewpoint the mathematics behind Enigma is fairly elementary, it turned out to be crucial for the future development of computer science.”

Written by Emily Nunez

The College of Computer, Mathematical, and Natural Sciences hosted a Reddit Ask-Me-Anything spotlighting epidemic modeling research.

University of Maryland Mathematics Professor Abba Gumel participated in an Ask-Me-Anything (AMA) user-led discussion on Reddit to answer questions about the mathematics of infectious diseases on April 9, 2025.

Gumel’s research group develops and analyzes novel mathematical models for gaining insight and understanding of the transmission dynamics and control of emerging and re-emerging infectious diseases of major public/global health significance. Members of Gumel’s lab joined him to answer questions, including postdoctoral researchers Alex Safsten and Arnaja Mitra and visiting assistant research scientist Salihu Musa.

This Reddit AMA has been edited for length and clarity.

My understanding is that diseases tend to spread exponentially in their initial stages. Is this accurate, and/or is it contingent on the disease itself? At what point, if any, does this trend begin to flatten?

(Gumel) That's a great question. Yes, diseases generally spread exponentially during the early stages (particularly if the reproduction number of the disease is greater than 1), and begin to decline as interventions and mitigation measures are implemented or people change their behavior. Most diseases tend to have a single peak and decline to lower or elimination levels with time and as the population of susceptible individuals decreases. 

Unfortunately, pandemics of influenza-like illnesses do not have single peaks; they have multiple peaks driven by factors such as human behavior, emergence of new variants, inadequate control resources, and so on.

How integral is an education in bioinformatics/computer science in biological science as it currently stands?

(Safsten) I find the ability to write programs, solve equations that arise in my models of biological problems very helpful. The models I tend to work on have unusual features that cannot be handled by standard equation-solving packages, so I develop my own algorithms for solving these models.

(Gumel) Students of biological sciences should be well-versed in computational and data analysis tools needed for studying the biological systems of interest.

(Musa) Bioinformatics and computer science are now key to biology. Analyzing large-scale data such as genomics, protein structures or disease patterns requires coding, algorithms and data science tools.

If someone were interested in the cross-section of math and biology, what recommendations would you give them to work in the field in the future?

(Safsten) Most people who work in mathematical biology are either very applied mathematicians or very theoretical biologists. I am the former, so I can mostly speak to that career path. I recommend studying differential equations, dynamical systems and linear algebra. You'll also benefit from some programming experience. Then, look into whatever areas of biology interest you. Whatever areas you choose, you will find unanswered questions that can be addressed with mathematical analysis.

(Mitra) If someone is interested in math and biology—especially if looking to switch into the field of epidemiology, systems biology, population dynamics or computational biology—besides having a math background, it will be good to consider some introductory biology, genetics and ecology. Also, if someone has no prior experience in programming languages, you can start with MATLAB or Python for simulation and data analysis. And if you want to do a simple computation, you may choose Mathematica or Maple. If you want to do some statistics modeling, then it's good to have some basic statistics or parameter estimation knowledge.

(Gumel) You're making a good choice to dabble in the beautiful world of mathematics and biology! That's where the real action is. The synergy between mathematics and biology provides exciting new challenges to mathematicians, sometimes requiring the development of new mathematical tools and branches (such as topological data analysis, uncertainty quantification and even nowadays, machine learning and AI tools). In general, to be successful within the space of mathematical biology, one has to have a deep appreciation for both mathematics, biology and all the other tools that are needed to succeed, including statistics, optimization, computation, data analytics, etc. 

One also has to have the desire to learn—for example, if you're a mathematician, you have to have the desire and capacity to learn the basic tools in biology to design, analyze and simulate realistic mathematical models for the biological phenomenon being modeled. Likewise, a biologist or someone in other sciences interested in doing modeling should also be comfortable learning the basic mathematical, statistical and computational tools needed to model the phenomenon. You can start with some of the classical literature on the topic, such as this Kermack-McKendrick 1927 paper and Hethcote's Scientific American review.

What is your understanding of the connection between pandemics and their frequency? Is there math backing up why we had so much time between the Spanish flu and the COVID-19 outbreak?

(Gumel) The 1918-1919 influenza pandemic was caused by the H1N1 influenza A virus. We have seen multiple outbreaks of the H1N1 pandemic since the 1918 pandemic, including the 2009 H1N1 swine flu pandemic, which started in Mexico and the United States. On the other hand, COVID-19 was caused by a coronavirus biologically similar to two previous coronavirus pandemics (the SARS pandemic of 2002 and the MERS pandemic of 2012).

Pandemics are generally consequences of spillover events from animals to humans. The frequency depends on the level of interaction between animals and humans. As long as humans continue to encroach on natural habitats of animals and alter or act in ways that affect the natural environment, we are constantly a mutation or two away from a spillover that could lead to a pandemic in humans. Sadly, it's a question of when, not if, we will be hit with the next pandemic (especially of respiratory pathogens).

(Safsten) There is no regular cycle for pandemics, but models show that globalization, urbanization, human action (climate change, land-use changes, etc.) and zoonotic spillovers increase the risk of pandemics.

In hindsight, was the two-week “quarantine to stop the spread” viable, mathematically? What percentage of the U.S. population would’ve needed to cooperate?

(Gumel) It's true that quarantine of symptomatic people for two weeks (away from interaction with the general population) will be useful in curtailing the spread of the disease, since the two-week quarantine period matches the incubation period of the disease. 

The big problem with COVID is that many of the transmitters are asymptomatic and have no idea they have the disease. Therefore, they are not in quarantine. In that sense, COVID-19 is different from other diseases where the main transmitters are people with symptoms. Because of that, quarantine and isolation alone are not sufficient to effectively mitigate or control the disease. For COVID, we needed a hybrid strategy that involved quarantine, large-scale testing, contact tracing, use of face masks and pharmaceutical interventions such as the vaccine and monoclonal antibody treatments.

Beyond explaining the mathematics behind it all, what are the most challenging obstacles you face when communicating your findings to public health professionals/policymakers?

(Gumel) Mathematics doesn't always have all the answers. Models are built based on well-thought-out assumptions, and predictions are subject to all sorts of uncertainties in the data, the assumptions, etc. It's very difficult to communicate these facts to public health professionals who are expecting actionable, day-to-day predictions. One of the things that seems to be missing in the modeling/science curriculum in general is how to effectively communicate our results and outputs of our modeling work to the general population. COVID-19 has highlighted the need for incorporating effective communications into science curricula in general, and the modeling of infectious diseases in particular.

What type of mathematical operations and theorems are you using for this research?

(Gumel) We generally design, calibrate, analyze and simulate various types of models (mechanistic/compartmental, network, statistical and some use AI/ML and agent-based models) to study the transmission dynamics and control of infectious diseases. We develop and use tools for nonlinear dynamical systems and other branches of mathematics to study the asymptotic properties of the steady-state solutions of the model, and characterize the bifurcation types (these allow us to obtain important epidemiological thresholds that are associated with the control or persistence of the disease in a population (such as the basic reproduction number and herd immunity thresholds). We also use statistical and optimization tools to fit models to data (and to estimate unknown parameters) and conduct uncertainty quantification and sensitivity analysis. Specifically, we use tools like Latin Hypercube Sampling and Partial Rank Correlation Coefficients to carry out global uncertainty and sensitivity analysis. Finally, we use these tools to determine optimal solutions, particularly when control resources are limited.

(Safsten) The models we use typically take the form of deterministic or stochastic systems of nonlinear differential equations that could be ordinary or partial (where the models have several other independent variables in addition to time). In the case of partial differential equations (PDEs), the models often take the form of semi-linear parabolic equations for which there are many analytical tools for analyzing the existence, uniqueness, boundedness and asymptotic stability of solutions. When external factors, such as climate change, behavior change, and gradual refinement of interventions, affect the system in a time-dependent way, the resulting models are non-autonomous. And there are very few theoretical tools for analyzing these models (for special cases, for instance, where the time-dependent parameters are periodic), thereby providing ample opportunities for aspiring graduate students to consider for their dissertations.

What are the most surprising factors that you found for diseases spreading?

(Mitra) One of the factors I found in my ongoing research is that the nonlinear effect of human behavior changes in one age group can significantly affect the transmission dynamics in another. For instance, even a tiny shift in behavior, such as reducing bednet use among children or a decrease in vaccine uptake, can lead to a disproportionate increase in disease transmission at the population level. Moreover, maturation can create a shift in the susceptible population, spatially in models where vaccine-induced immunity wanes over time.

(Gumel) Most human diseases are zoonotic diseases that jump from animal populations to humans; we humans are responsible for most of these diseases based on our actions that affect the natural habitats and dynamics of nonhuman primates. Understanding the One Health approach to public health—where public health is viewed holistically from the point of view of nonhuman primates, humans and the environment—is so critical to improving human health.

The other surprising thing is the role of the asymptomatic and presymptomatic transmission in the spread and control of COVID-19 (before COVID, diseases were mostly transmitted by people with clinical symptoms, not largely by those without symptoms).

While some diseases are controllable using basic public health measures such as quarantine, isolation and hand-washing (e.g., SARS of 2002-2003 and even MERS of 2012), others require the use of both non-pharmaceutical and pharmaceutical interventions (e.g., COVID-19).

(Musa) In addition to asymptomatic transmission of infectious diseases (such as COVID-19), there were also superspreading events where a small number of individuals affected an unusually large number of others. See our paper on modeling superspreading of COVID. This dynamic made surveillance, contact tracing and control of COVID-19 more difficult.